Theorem en2top 20789

Description: If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
en2top	|- ( J e. (TopOn ` X ) -> ( J ~~ 2o <-> ( J = { (/) , X } /\ X =/= (/) ) ) )

Proof of Theorem en2top

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> J ~~ 2o )
2		toponss 20731	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` X ) /\ x e. J ) -> x C_ X )
3	2	ad2ant2rl 785	. . . . . . . . . . . . . . . . 17 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ ( X = (/) /\ x e. J ) ) -> x C_ X )
4		simprl 794	. . . . . . . . . . . . . . . . 17 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ ( X = (/) /\ x e. J ) ) -> X = (/) )
5		sseq0 3975	. . . . . . . . . . . . . . . . 17 |- ( ( x C_ X /\ X = (/) ) -> x = (/) )
6	3, 4, 5	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ ( X = (/) /\ x e. J ) ) -> x = (/) )
7		velsn 4193	. . . . . . . . . . . . . . . 16 |- ( x e. { (/) } <-> x = (/) )
8	6, 7	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ ( X = (/) /\ x e. J ) ) -> x e. { (/) } )
9	8	expr 643	. . . . . . . . . . . . . 14 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> ( x e. J -> x e. { (/) } ) )
10	9	ssrdv 3609	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> J C_ { (/) } )
11		topontop 20718	. . . . . . . . . . . . . . . 16 |- ( J e. (TopOn ` X ) -> J e. Top )
12		0opn 20709	. . . . . . . . . . . . . . . 16 |- ( J e. Top -> (/) e. J )
13	11, 12	syl 17	. . . . . . . . . . . . . . 15 |- ( J e. (TopOn ` X ) -> (/) e. J )
14	13	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> (/) e. J )
15	14	snssd 4340	. . . . . . . . . . . . 13 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> { (/) } C_ J )
16	10, 15	eqssd 3620	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> J = { (/) } )
17		0ex 4790	. . . . . . . . . . . . 13 |- (/) e. _V
18	17	ensn1 8020	. . . . . . . . . . . 12 |- { (/) } ~~ 1o
19	16, 18	syl6eqbr 4692	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> J ~~ 1o )
20	19	olcd 408	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> ( J = (/) \/ J ~~ 1o ) )
21		sdom2en01 9124	. . . . . . . . . 10 |- ( J ~< 2o <-> ( J = (/) \/ J ~~ 1o ) )
22	20, 21	sylibr 224	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> J ~< 2o )
23		sdomnen 7984	. . . . . . . . 9 |- ( J ~< 2o -> -. J ~~ 2o )
24	22, 23	syl 17	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) /\ X = (/) ) -> -. J ~~ 2o )
25	24	ex 450	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> ( X = (/) -> -. J ~~ 2o ) )
26	25	necon2ad 2809	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> ( J ~~ 2o -> X =/= (/) ) )
27	1, 26	mpd 15	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> X =/= (/) )
28	27	necomd 2849	. . . 4 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> (/) =/= X )
29	13	adantr 481	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> (/) e. J )
30		toponmax 20730	. . . . . 6 |- ( J e. (TopOn ` X ) -> X e. J )
31	30	adantr 481	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> X e. J )
32		en2eqpr 8830	. . . . 5 |- ( ( J ~~ 2o /\ (/) e. J /\ X e. J ) -> ( (/) =/= X -> J = { (/) , X } ) )
33	1, 29, 31, 32	syl3anc 1326	. . . 4 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> ( (/) =/= X -> J = { (/) , X } ) )
34	28, 33	mpd 15	. . 3 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> J = { (/) , X } )
35	34, 27	jca 554	. 2 |- ( ( J e. (TopOn ` X ) /\ J ~~ 2o ) -> ( J = { (/) , X } /\ X =/= (/) ) )
36		simprl 794	. . 3 |- ( ( J e. (TopOn ` X ) /\ ( J = { (/) , X } /\ X =/= (/) ) ) -> J = { (/) , X } )
37	17	a1i 11	. . . 4 |- ( ( J e. (TopOn ` X ) /\ ( J = { (/) , X } /\ X =/= (/) ) ) -> (/) e. _V )
38	30	adantr 481	. . . 4 |- ( ( J e. (TopOn ` X ) /\ ( J = { (/) , X } /\ X =/= (/) ) ) -> X e. J )
39		simprr 796	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ ( J = { (/) , X } /\ X =/= (/) ) ) -> X =/= (/) )
40	39	necomd 2849	. . . 4 |- ( ( J e. (TopOn ` X ) /\ ( J = { (/) , X } /\ X =/= (/) ) ) -> (/) =/= X )
41		pr2nelem 8827	. . . 4 |- ( ( (/) e. _V /\ X e. J /\ (/) =/= X ) -> { (/) , X } ~~ 2o )
42	37, 38, 40, 41	syl3anc 1326	. . 3 |- ( ( J e. (TopOn ` X ) /\ ( J = { (/) , X } /\ X =/= (/) ) ) -> { (/) , X } ~~ 2o )
43	36, 42	eqbrtrd 4675	. 2 |- ( ( J e. (TopOn ` X ) /\ ( J = { (/) , X } /\ X =/= (/) ) ) -> J ~~ 2o )
44	35, 43	impbida 877	1 |- ( J e. (TopOn ` X ) -> ( J ~~ 2o <-> ( J = { (/) , X } /\ X =/= (/) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   class class class wbr 4653   ` cfv 5888   1oc1o 7553   2oc2o 7554   ~~ cen 7952   ~< csdm 7954   Topctop 20698  TopOnctopon 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-top 20699  df-topon 20716

This theorem is referenced by:  hmphindis  21600